Question: All of the 5th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$2.00$ each for students, and the group paid $$29.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$8.50$ each for students, and the group paid $$119.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2y = 29}$ ${34x+8.5y = 119}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-8y = -116}$ ${34x+8.5y = 119}$ Add the top and bottom equations together. $ 0.5y = 3 $ $ y = \dfrac{3}{0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+2y = 29}$ to find $x$ ${8.5x + 2}{(6)}{= 29}$ $8.5x+12 = 29$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {34x+8.5y = 119}$ and get the same answer for $x$ ${34x + 8.5}{(6)}{= 119}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.